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previous case, we can write where Hence, for z # 0, f(z+h)-f(z) -ΣnCnz"-1 =ΣCnbn, h n=0 n=2 If z = 0, bn =h"-1 and the proof follows easily. Otherwise, to obtain a us estimate for bn, we must be a little more careful. Note then that (3) = ¹² „n²|h| |z1² |bn|≤ Σ 2141 bn n(n-1)(n-k+1) k! n n • Σ (²) hk-1₂n-k k=2 k=2 n - < ₁ ² ( x ²² 2₂) n (²2) 14-1²-²212²-(²-2) k-2 for k ≥ 2.

previous case, we can write where Hence, for z # 0, f(z+h)-f(z) -ΣnCnz"-1 =ΣCnbn, h n=0 n=2 If z = 0, bn =h"-1 and the proof follows easily. Otherwise, to obtain a us estimate for bn, we must be a little more careful. Note then that (3) = ¹² „n²|h| |z1² |bn|≤ Σ 2141 bn n(n-1)(n-k+1) k! n n • Σ (²) hk-1₂n-k k=2 k=2 n - < ₁ ² ( x ²² 2₂) n (²2) 14-1²-²212²-(²-2) k-2 for k ≥ 2.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 13E
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Why does the inequality highlighted in red occur?

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Case (2): 0 < R <∞0. Let |z| = R-28, d > 0, and assume |h| <d. Then [z + h| < R and, as in the previous case. we can write where (3) = Hence, for z # 0, f(z+h)-f(z) h VI If z = 0, bn = h"-1 and the proof follows easily. Otherwise, to obtain a useful estimate for b, we must be a little more careful. Note then that = 5 bn ‚n²|h|· |z|² - ΣnCnz"-1 = [Cnbn, n=0 n=2 n = = Σ(^) n²-¹_n-k. k=2 n(n-1)(n-k+1) k! = n²|h| |z|² n²|h| |z|² n |bn|≤ -2 (₁-²-2) ||h|k−2|z|n—(k−2) k=2 ²H()W=-/ -(|z|+|h|)" -(R-8)" n m² (₁ ²₂2) for <n² for k ≥ 2.
Transcribed Image Text:Case (2): 0 < R <∞0. Let |z| = R-28, d > 0, and assume |h| <d. Then [z + h| < R and, as in the previous case. we can write where (3) = Hence, for z # 0, f(z+h)-f(z) h VI If z = 0, bn = h"-1 and the proof follows easily. Otherwise, to obtain a useful estimate for b, we must be a little more careful. Note then that = 5 bn ‚n²|h|· |z|² - ΣnCnz"-1 = [Cnbn, n=0 n=2 n = = Σ(^) n²-¹_n-k. k=2 n(n-1)(n-k+1) k! = n²|h| |z|² n²|h| |z|² n |bn|≤ -2 (₁-²-2) ||h|k−2|z|n—(k−2) k=2 ²H()W=-/ -(|z|+|h|)" -(R-8)" n m² (₁ ²₂2) for <n² for k ≥ 2.
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